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Theorem iunsuc 4310
 Description: Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
Hypotheses
Ref Expression
iunsuc.1
iunsuc.2
Assertion
Ref Expression
iunsuc
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem iunsuc
StepHypRef Expression
1 df-suc 4261 . . 3
2 iuneq1 3794 . . 3
31, 2ax-mp 5 . 2
4 iunxun 3860 . 2
5 iunsuc.1 . . . 4
6 iunsuc.2 . . . 4
75, 6iunxsn 3857 . . 3
87uneq2i 3195 . 2
93, 4, 83eqtri 2140 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1314   wcel 1463  cvv 2658   cun 3037  csn 3495  ciun 3781   csuc 4255 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-iun 3783  df-suc 4261 This theorem is referenced by: (None)
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